\newproblem{lay:4_7_9}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.7.9}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $B=\{\mathbf{b}_1,\mathbf{b}_2\}$ and $C=\{\mathbf{c}_1,\mathbf{c}_2\}$ be bases for a vector space $\mathbb{R}^2$ with
	$\mathbf{b}_1=\begin{pmatrix}7\\5\end{pmatrix}$, $\mathbf{b}_2=\begin{pmatrix}-3\\-1\end{pmatrix}$, $\mathbf{c}_1=\begin{pmatrix}1\\-5\end{pmatrix}$, and
	$\mathbf{c}_2=\begin{pmatrix}-2\\2\end{pmatrix}$. Find the change-of-coordinates matrices from $B$ to $C$ and from $C$ to $B$
}{
  % Solution
	The change-of-coordinates matrices from $B$ and $C$ to the standard basis of $\mathbb{R}^2$ are given by
	\begin{center}
		$P_{E\leftarrow B}=\begin{pmatrix}[\mathbf{b}_1]_E & [\mathbf{b}_2]_E\end{pmatrix}=\begin{pmatrix}7 & -3 \\ 5 & -1\end{pmatrix}$ \\
		$P_{E\leftarrow C}=\begin{pmatrix}[\mathbf{c}_1]_E & [\mathbf{c}_2]_E\end{pmatrix}=\begin{pmatrix}1 & -2 \\ -5 & 2\end{pmatrix}$ \\
	\end{center}
	Now we note that for each one of the basis we have
	\begin{center}
		$[\mathbf{x}]_E=P_{E\leftarrow B}[\mathbf{x}]_B=P_{E\leftarrow C}[\mathbf{x}]_C \Rightarrow [\mathbf{x}]_C=P_{E\leftarrow C}^{-1}P_{E\leftarrow B}[\mathbf{x}]_B$ \\
	\end{center}
	In this particular case
	\begin{center}
		$P_{C\leftarrow B}=P_{E\leftarrow C}^{-1}P_{E\leftarrow B}=\begin{pmatrix}1 & -2 \\ -5 & 2\end{pmatrix}^{-1}\begin{pmatrix}7 & -3 \\ 5 & -1\end{pmatrix}=
		   \begin{pmatrix}-3& 1 \\ -5 & 2\end{pmatrix}$
	\end{center}
	In the other direction
	\begin{center}
		$P_{B\leftarrow C}=P_{C\leftarrow B}^{-1}=\begin{pmatrix}-2& 1 \\ -5 & 3\end{pmatrix}$
	\end{center}
}
\useproblem{lay:4_7_9}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
